In two recent studies, Dudok de Wit et al. (2005) and Kretzschmar et al. (2006) have shown that the solar Ultra-Violet spectrum between 25 and 195 nm can be reconstructed from the observation of a set of 6 to 10 carefully chosen spectral lines. The best set of lines, however, is application dependent. In this study, we demonstrate that a good candidate for aeronomy applications consists of the following 6 lines: H I at 102.572 nm, C III at 97.702 nm, O V at 62.973 nm, He I at 58.433 nm, Fe XV at 28.415 nm and He II at 30.378 nm. The TRANSCAR model is used to quantify the impact of each individual line on the density, temperature and velocity profiles. Using a multidimensional scaling technique, we show how to select from this the best set of lines. Although this selection is motivated by the specification of the ionosphere, our set of lines is also found to be appropriate for reconstructing the variability of the solar spectrum between 25 and 195 nm.

The aim of the paper is to study the link between non-additivity of some entropies and their boundedness. We propose an axiomatic construction of the entropy relying on the fact that entropy belongs to a group isomorphic to the usual additive group. This allows to show that the entropies that are additive with respect to the addition of the group for independent random variables are nonlinear transforms of the Rényi entropies, including the particular case of the Shannon entropy. As a particular example, we study as a group a bounded interval in which the addition is a generalization of the addition of velocities in special relativity. We show that Tsallis–Havrda–Charvat entropy is included in the family of entropies we define. Finally, a link is made between the approach developed in the paper and the theory of deformed logarithms.

The goal of the paper is the study of suboptimal quantizer based detectors. We place ourselves in the situation where internal noise is present in the hard implementation of the thresholds. We hence focus on the study of random quantizers, showing that they present the noise-enhanced detection property. The random quantizers studied are of two types: time invariant when sampled once for all the observations, time variant when sampled at each time. They are built by adding fluctuations on the thresholds of a uniform quantizer. If the uniform quantizer is matched to the symmetry of the detection problem, adding fluctuation deteriorates the performance. If the uniform quantizer is mismatched, adding noise can improve the performance. Furthermore, we show that the time varying quantizer is better than the time invariant quantizer, and we show that both are more robust than the optimal quantizer. Finally, we introduce the adapted random quantizer for which the levels are chosen in order to approximate the likelihood ratio.

The solar EUV irradiance is a key input for thermospheric and ionospheric models. Difficulties in continuously measuring the calibrated spectrum has prompted the use of various surrogate quantities. Although most proxies correlate quite well with the spectral variability, their use for modelling purposes becomes increasingly unsatisfactory. A different and data-driven approach is considered here, in which the EUV spectrum is reconstructed from a linear combination of a few, calibrated and carefully selected spectral lines. This approach is based on a statistical analysis of the temporal variability of EUV spectra, as recorded by the TIMED satellite. A basic set of lines is extracted, from which the salient features of the spectral variability can be reconstructed. The best results are achieved with a selection of 5 to 8 of these lines. This study focuses on the methodology for selecting these lines, which can also be used for instrument specification and provides new insight into the comparison of solar proxies against the EUV irradiance.

In this paper, we revisit the problem of detecting a known signal corrupted by an independent identically distributed α-stable noise. The implementation of the optimal receiver, i.e. the log-likelihood ratio, requires the explicit expression of the probability density function of the noise. In the general α-stable case, there exists no closed-form for the probability density function of the noise. To avoid the numerical evaluation of the probability density function of the noise, we propose to study a parametric suboptimal detector based on properties of α-stable noise and on implementation considerations. We focus our attention on several optimization criteria of the parameters, showing that our choice allows the optimization without using the explicit expression of the noise probability density function. The chosen detector allows to retrieve the optimal Gaussian detector (matched filter) as well as the locally optimal detector in the Cauchy context. The performance of the detector is studied and compared to usual detectors and to the optimal detector. The robustness of the detector against the signal amplitude and the stability index of the noise is discussed.

We present a theory of stochastic processes that are finite size scale invariant. Such processes are invariant under generalized dilations that operate on bounded ranges of scales and amplitudes. We recall here the theory of deterministic finite size scale invariance, and introduce an operator called Lamperti transform that makes equivalent generalized dilations and translations. This operator is then used to defined finite size scale invariant processes as image of stationary processes. The example of the Brownian motion is presented is some details to illustrate the definitions. We further extend the theory to the case of finite size scale invariant processes with stationary increments.

The paper proposes a sequential Monte Carlo simulation method for equalizing a satellite communication channel. The main difficulties encountered are the nonlinear distortions due to the amplifier stage in the satellite. The aim of the method is to restore blindly the emitted message by considering a Bayesian approach. Thus, prior knowledge on the modeling of the nonlinearity is taken into account in the posterior distribution of the input sequence. Such a distribution is very difficult to study and thus motivates the implementation of Monte Carlo techniques. This approach makes it possible to solve the problem for a simplified model. The simulation scheme dealing with the complete transmission chain uses the method developed for the simplified model. Performance of the equalization algorithm is evaluated using bit error rate versus signal-to-noise ratio curves.

The aim of the paper is to show that the nonlinear effect known as stochastic resonance, which corresponds to the improvement of the processing of information by noise, occurs naturally in some detection problems. We illustrate this by studying the problem of detecting a small amplitude sinusoid in non-Gaussian noise. We show that in some cases, the nonlinearity that appears in locally optimal detectors can be viewed as a stochastic resonator. If the parameters of the locally optimal detector (LOD) are not well tuned, the performance can be improved by the addition of noise.

The Lamperti transformation defines a one-to-one correspondence between stationary processes on the real line and self-similar processes on the real half-line. Although dating back to 1962, this fundamental result has further received little attention until a recent past, and it is the purpose of this chapter to survey the Lamperti transformation and its (effective and/or potential) applications, with emphasis on variations which can be made on the initial formulation. After having recalled basics of the transform itself, some results from the literature will be reviewed, which can be broadly classified in two types. In a first category, classical concepts from stationary processes and linear filtering theory, such as linear time-invariant systems or ARMA modeling, can be given self-similar counterparts by a proper "lampertization" whereas, in a second category, problems such as spectral analysis or prediction of self-similar processes can be addressed with classical tools after stationarization by a converse "delampertization". Variations and new results will then be discussed by investigating consequences of the Lamperti transformation when applied to weakened forms of stationarity, and hence of self-similarity. Different forms of locally stationary processes will be considered this way, as well as cyclostationary processes for which "lampertization" will be shown to offer a suitable framework for defining a stochastic extension to the notion of discrete scale invariance which has recently been put forward as a central concept in many critical systems. Issues concerning the practical analysis (and synthesis) of such processes will be examined, with a possible use of Mellin-based tools operating directly in the space of scaling data.

This paper deals with the use of stochastic resonance (SR) for detection purposes. The nonlinear physical phenomenon of SR generally occurs in dynamical bistable systems excited by a noisy sine: such systems are able to force cooperation between sine and noise such that the noise amplifies the sine. Because of this non-intuitive effect, the use of SR can be envisaged to detect small amplitude sines corrupted by additive noise. In this paper we recall some basics of detection and then show why SR can be used in sine detection context. After recalling some basics of SR in discrete time, we show how to use SR in a detection scheme.

A definition of stochastic discrete scale invariance (DSI) is proposed and its properties studied. It is shown how the Lamperti (1962) transformation, which transforms stationarity in self-similarity, is also a means to connect processes deviating from stationarity and processes which are not exactly scale invariant: in particular we interpret DSI as the image of cyclostationarity. This theoretical result is employed to introduce a multiplicative spectral representation of DSI processes based on the Mellin transform, and preliminary remarks are given about estimation issues.

This paper studies the problem of tracking a Brownian phase with linear drift observed to within one digital modulation and one additive white Gaussian noise. This problem is of great importance as it models the problem of carrier synchronization in digital communications. The ultimate performances achievable for this problem are evaluated and are compared to the performances of three solutions of the problem. The optimal filter cannot be explicitly calculated and one goal of the paper is to implement it using recent sequential Monte-Carlo techniques known as particle filtering. This approach is compared to more traditional loops such as the Costas loop and the decision feedback loop. Moreover, since the phase has a linear drift, the loops considered are second-order loops. To make fair comparisons, we exploit all the known information to put the loops in their best configurations (optimal step sizes of the loops). We show that asymptotically, the loops and the particle filter are equivalent in terms of mean square error. However, using Monte-Carlo simulations we show that the particle filter outperforms the loops when considering the mean acquisition time (convergence rate), and we argue that the particle filter is also better than the loops when dealing with the important problem of mean time between cycle slips.

In this paper, we present a detector of transient acoustic signals that combines two powerful detection tools: a local wavelet analysis and higher-order statistical properties of the signals. The use of both techniques makes detection possible in low signal-to-noise ratio conditions, when other means of detection are no longer sufficient. The proposed algorithm uses the adapted wavelet packet transform. It leads to a partition of the signal which is ‘optimal’ according to a criterion that tests the Gaussian nature of the frequency bands. To get a time dependent detection curve, we perform a de-noising procedure on the wavelet coefficients: The Gaussian coefficients are set to zero. We then apply a classical method of detection on the time reconstructed de-noised signal. We study the performance of the detector in terms of experimental ROC curves. We show that the detector performs better than decompositions using other classical splitting criteria. In the last part, we present an application of the algorithm on real flow recordings of nuclear plant pipings. The detector indicates the presence of a missing body in the piping at some instants not seen with a classical energy detector.

Source separation consists in recovering signals mixed by an unknown transmission channel. Likelihood and information theory or higher order statistics can be used to perform the separation. This paper proposes a Bayesian approach to the problem of an instantaneous linear mixing, considering the source signals are discrete valued. The Bayesian inference enables to take in account jointly prior information and the information available on the observation signals. This approach implies complex calculations which can be achieved through Monte Carlo Markov Chain (MCMC) simulation methods. The separation method for binary inputs was exposed in and is now extended to PSK source signals.

We define and study stochastic discrete scale invariance (DSI), a property which requires invariance by dilation for certain preferred scaling factors only. We prove that the Lamperti transformation, known to map self-similar processes to stationary processes, is an important tool to study these processes and gives a more general connection: in particular between DSI and cyclostationarity. Some general properties of DSI processes are given. Examples of random sequences with DSI are then constructed and illustrated. We address finally the problem of analysis of DSI processes, first using the inverse Lamperti( 1962) transformation to analyse DSI processes by means of cyclostationary methods. Second we propose to re-write these tools directly in a Mellin formalism.

The aim of the paper is to provide a discussion on the notion of nonlinear interaction between scales. This notion is extremely important in the study of nonlinear physical phenomena, such as the study of the energy cascade in fully developed turbulence in fluids. The discussion is centered on the temptative definition of higher-order time-multiscale distributions. A simple answer which we call the biscalogram enlightens the main difficulty arising in our attempt to define nonlinear interactions between scales. Some simulations illustrate that point. Furthermore, two different classes of higher-order time-multiscale distributions are proposed, the first one based on a set of affine covariance properties, the other on a set of invariance-covariance properties

This paper is devoted to the detection of phase- or frequency-modulated signals corrupted by the reverberation noise they have created. When the reverberation is modeled as a time-varying stochastic filtering of the emitted signal, the optimal detector consists of prewhitening the observation before applying a matched filter. This theory is very difficult to implement because of the nonstationarity of the reverberation noise. In this paper, a suboptimal detector proposed by Kay and Salisbury in 1990 is generalized to phase or frequency modulated signals. The reverberation is assumed locally stationary; the signal is cut into blocks and the whitening is performed as follows: An autoregressive (AR) modeling of one block allows the whitening of the following block. After whitening, an appropriate matched filter is applied. This scheme is explained in detail. The local stationarity assumption of the reverberation noise is discussed in light of the Itakura spectral distance. Performance of the proposed approach is evaluated theoretically when all parameters are known, and experimentally using three different real-data sets.

This paper intends to show how the theory of stochastic cyclostationary processes can be used to study stochastic resonance in static nonlinearities. The statistic we use is the covariance function of the output. The covariance is a second-order cumulant and is not dependent on by the mean. Furthermore, this covariance is not averaged in time as is usually done in the stochastic resonance literature. A two-dimensional Fourier transform of the covariance gives the so-called spectral correlation. The spectral correlation depends on the usual harmonic frequency and on another frequency, called cycle frequency. The cyclostationarity of a signal makes the spectral correlation discrete in the cycle frequency. The zero cycle frequency corresponds to the usual "stationary power spectrum" used in the stochastic resonance literature. We thus exploit all the second-order statistical information. We first revisit classical stochastic resonance in threshold devices using the spectral correlation, showing that the effect is seen for nonzero cycle frequencies. The cases of additive and multiplicative noise are detailed. We then study stochastic resonance in threshold devices for communication signals. These signals are usually modeled as stochastic cyclostationary processes. We show that stochastic resonance occurs, and the phenomenon is quantified using the spectral correlation of the output: The amplitude of the spectral correlation at nonzero cycle frequencies presents a maximum as the power of the input noise is increased.

This paper deals with stochastic resonance. This nonlinear physical phenomenon generally occurs in bistable systems excited by random input noise plus a sinusoid. Through its internal dynamics, such a system forces cooperation between the input noise and the input sine: provided the existence of fine tuning between the power noise and the dynamics, the system reacts periodically at the frequency of the sine. Of particular interest is the fact that the local output signal-to-noise ratio presents a maximum when plotted against the input noise power; the system resounds stochastically. Continuous-time systems have already been studied. We study the ability of intrinsically discrete-time systems [general nonlinear AR(1) models] to produce stochastic resonance. It is then suggested that such discrete systems can be used in signal processing.